Gauss divergence application correct or not

Question

Let $W$ be the region inside the solid cylinder $x^2+y^2\leq 4$ between the planes $z=0$ and the paraboloid $z=x^2+y^2$. Let $S$ be the boundary of $W$. Evaluate$$\int\int_S\vec{F}.\hat{n}dS$$ where $\vec{F}=(x^2+y^2-4)\hat{i}+3xy\hat{j}+(2xz+z^2)\hat{k}$ and $\hat{n}$ is the outward normal. I calculate the divergence of $F$ i.e $div(\vec{F})=7x+2z$ and used the Gauss Divergence theorem so i got $$\int\int_{Circle\;Radius\leq 2}\int_{0}^{x^2+y^2}7x+2z \;dzdxdy$$ is my approach right. can anybody explain if it is right why can we apply gauss divergence theorem, i still can't convince myself